Left Termination of the query pattern delete_in_3(a, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

delete(X, tree(X, void, Right), Right).
delete(X, tree(X, Left, void), Left).
delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
delmin(tree(Y, void, Right), Y, Right).
delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
less(0, s(X3)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

delete(a,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
delete_in: (f,b,f) (b,b,f)
delmin_in: (b,f,f)
less_in: (f,b) (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x2, x3, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x2, x3, x4, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x2, x3, x4, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x2, x3, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x2, x3, x4, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x2, x3)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x2, x3, x4, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x1, x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(20) TRUE

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
  The graph contains the following edges 1 > 1

(27) TRUE

(28) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(29) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(30) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(31) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(33) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)
  The graph contains the following edges 1 > 1

(34) TRUE

(35) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(36) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(37) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(38) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Y, Left, Right, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left)
DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Y, Left, Right, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The set Q consists of the following terms:

less_in_gg(x0, x1)
U7_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(40) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x2, x3, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x1, x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x1, x2, x3, x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x2, x3, x4, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1, x2)
U7_ag(x1, x2, x3)  =  U7_ag(x2, x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x2, x3, x4, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x1, x2, x3, x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg(x1, x2)
U7_gg(x1, x2, x3)  =  U7_gg(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x4, x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x1, x2, x3, x4, x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x2, x3, x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x2, x3, x4, x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(41) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(42) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga(x1)
U7_ga(x1, x2, x3)  =  U7_ga(x1, x3)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x2, x3, x4, x6)

We have to consider all (P,R,Pi)-chains

(43) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
delete_in: (f,b,f) (b,b,f)
delmin_in: (b,f,f)
less_in: (f,b) (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(44) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)

(45) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)

We have to consider all (P,R,Pi)-chains

(46) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_AGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_AGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_GAA(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U7_AG(X, Y, less_in_ag(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_AGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_GGA(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
DELETE_IN_GGA(X, tree(X, Left, Right), tree(Y, Left, Right1)) → DELMIN_IN_GAA(Right, Y, Right1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U7_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_GGA(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GG(Y, X)
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_GGA(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U7_GA(X, Y, less_in_ga(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_AGA(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x1, x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x7)
U2_AGA(x1, x2, x3, x4, x5, x6)  =  U2_AGA(x3, x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)
U7_AG(x1, x2, x3)  =  U7_AG(x3)
U3_AGA(x1, x2, x3, x4, x5, x6)  =  U3_AGA(x1, x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x6)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U7_GA(x1, x2, x3)  =  U7_GA(x3)
U5_AGA(x1, x2, x3, x4, x5, x6)  =  U5_AGA(x6)

We have to consider all (P,R,Pi)-chains

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 18 less nodes.

(48) Complex Obligation (AND)

(49) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(50) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(51) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X), s(Y)) → LESS_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)

We have to consider all (P,R,Pi)-chains

(52) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(53) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_GA(s(X)) → LESS_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(54) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GA(s(X)) → LESS_IN_GA(X)
  The graph contains the following edges 1 > 1

(55) TRUE

(56) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_GG(x1, x2)  =  LESS_IN_GG(x1, x2)

We have to consider all (P,R,Pi)-chains

(57) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(58) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(59) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(60) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(61) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(62) TRUE

(63) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(64) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(65) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(X), s(Y)) → LESS_IN_AG(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

We have to consider all (P,R,Pi)-chains

(66) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(67) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(68) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• LESS_IN_AG(s(Y)) → LESS_IN_AG(Y)
  The graph contains the following edges 1 > 1

(69) TRUE

(70) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(71) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(72) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1), Y, tree(X, Left1, X2)) → DELMIN_IN_GAA(Left, Y, Left1)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
DELMIN_IN_GAA(x1, x2, x3)  =  DELMIN_IN_GAA(x1)

We have to consider all (P,R,Pi)-chains

(73) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(74) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(75) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• DELMIN_IN_GAA(tree(X, Left, X1)) → DELMIN_IN_GAA(Left)
  The graph contains the following edges 1 > 1

(76) TRUE

(77) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

We have to consider all (P,R,Pi)-chains

(78) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(79) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y))
U2_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → DELETE_IN_GGA(X, Left, Left1)
DELETE_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → DELETE_IN_GGA(X, Right, Right1)

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
DELETE_IN_GGA(x1, x2, x3)  =  DELETE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x3, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x4, x6)

We have to consider all (P,R,Pi)-chains

(80) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(81) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Left, less_in_gg(X, Y))
U2_GGA(X, Left, less_out_gg) → DELETE_IN_GGA(X, Left)
DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Right, less_in_gg(Y, X))
U4_GGA(X, Right, less_out_gg) → DELETE_IN_GGA(X, Right)

The TRS R consists of the following rules:

less_in_gg(0, s(X3)) → less_out_gg
less_in_gg(s(X), s(Y)) → U7_gg(less_in_gg(X, Y))
U7_gg(less_out_gg) → less_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
U7_gg(x0)

We have to consider all (P,Q,R)-chains.

(82) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• U2_GGA(X, Left, less_out_gg) → DELETE_IN_GGA(X, Left)
  The graph contains the following edges 1 >= 1, 2 >= 2

• U4_GGA(X, Right, less_out_gg) → DELETE_IN_GGA(X, Right)
  The graph contains the following edges 1 >= 1, 2 >= 2

• DELETE_IN_GGA(X, tree(Y, Left, Right)) → U2_GGA(X, Left, less_in_gg(X, Y))
  The graph contains the following edges 1 >= 1, 2 > 2

• DELETE_IN_GGA(X, tree(Y, Left, Right)) → U4_GGA(X, Right, less_in_gg(Y, X))
  The graph contains the following edges 1 >= 1, 2 > 2

(83) TRUE

(84) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

delete_in_aga(X, tree(X, void, Right), Right) → delete_out_aga(X, tree(X, void, Right), Right)
delete_in_aga(X, tree(X, Left, void), Left) → delete_out_aga(X, tree(X, Left, void), Left)
delete_in_aga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_aga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
delmin_in_gaa(tree(Y, void, Right), Y, Right) → delmin_out_gaa(tree(Y, void, Right), Y, Right)
delmin_in_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2)) → U6_gaa(X, Left, X1, Y, Left1, X2, delmin_in_gaa(Left, Y, Left1))
U6_gaa(X, Left, X1, Y, Left1, X2, delmin_out_gaa(Left, Y, Left1)) → delmin_out_gaa(tree(X, Left, X1), Y, tree(X, Left1, X2))
U1_aga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_aga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y))
less_in_ag(0, s(X3)) → less_out_ag(0, s(X3))
less_in_ag(s(X), s(Y)) → U7_ag(X, Y, less_in_ag(X, Y))
U7_ag(X, Y, less_out_ag(X, Y)) → less_out_ag(s(X), s(Y))
U2_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) → U3_aga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(X, void, Right), Right) → delete_out_gga(X, tree(X, void, Right), Right)
delete_in_gga(X, tree(X, Left, void), Left) → delete_out_gga(X, tree(X, Left, void), Left)
delete_in_gga(X, tree(X, Left, Right), tree(Y, Left, Right1)) → U1_gga(X, Left, Right, Y, Right1, delmin_in_gaa(Right, Y, Right1))
U1_gga(X, Left, Right, Y, Right1, delmin_out_gaa(Right, Y, Right1)) → delete_out_gga(X, tree(X, Left, Right), tree(Y, Left, Right1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) → U2_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y))
less_in_gg(0, s(X3)) → less_out_gg(0, s(X3))
less_in_gg(s(X), s(Y)) → U7_gg(X, Y, less_in_gg(X, Y))
U7_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U2_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) → U3_gga(X, Y, Left, Right, Left1, delete_in_gga(X, Left, Left1))
delete_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X))
U4_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) → U5_gga(X, Y, Left, Right, Right1, delete_in_gga(X, Right, Right1))
U5_gga(X, Y, Left, Right, Right1, delete_out_gga(X, Right, Right1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1))
U3_gga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
U3_aga(X, Y, Left, Right, Left1, delete_out_gga(X, Left, Left1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right))
delete_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X))
less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))
U4_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → U5_aga(X, Y, Left, Right, Right1, delete_in_aga(X, Right, Right1))
U5_aga(X, Y, Left, Right, Right1, delete_out_aga(X, Right, Right1)) → delete_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1))

The argument filtering Pi contains the following mapping:
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
void  =  void
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1)
U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x1, x6)
delmin_in_gaa(x1, x2, x3)  =  delmin_in_gaa(x1)
delmin_out_gaa(x1, x2, x3)  =  delmin_out_gaa(x2)
U6_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaa(x7)
U2_aga(x1, x2, x3, x4, x5, x6)  =  U2_aga(x3, x6)
less_in_ag(x1, x2)  =  less_in_ag(x2)
s(x1)  =  s(x1)
less_out_ag(x1, x2)  =  less_out_ag(x1)
U7_ag(x1, x2, x3)  =  U7_ag(x3)
U3_aga(x1, x2, x3, x4, x5, x6)  =  U3_aga(x1, x6)
delete_in_gga(x1, x2, x3)  =  delete_in_gga(x1, x2)
delete_out_gga(x1, x2, x3)  =  delete_out_gga
U1_gga(x1, x2, x3, x4, x5, x6)  =  U1_gga(x6)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x3, x6)
less_in_gg(x1, x2)  =  less_in_gg(x1, x2)
0  =  0
less_out_gg(x1, x2)  =  less_out_gg
U7_gg(x1, x2, x3)  =  U7_gg(x3)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x4, x6)
U5_gga(x1, x2, x3, x4, x5, x6)  =  U5_gga(x6)
U4_aga(x1, x2, x3, x4, x5, x6)  =  U4_aga(x4, x6)
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
U5_aga(x1, x2, x3, x4, x5, x6)  =  U5_aga(x6)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)

We have to consider all (P,R,Pi)-chains

(85) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(86) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) → U4_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X))
U4_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) → DELETE_IN_AGA(X, Right, Right1)

The TRS R consists of the following rules:

less_in_ga(0, s(X3)) → less_out_ga(0, s(X3))
less_in_ga(s(X), s(Y)) → U7_ga(X, Y, less_in_ga(X, Y))
U7_ga(X, Y, less_out_ga(X, Y)) → less_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
s(x1)  =  s(x1)
0  =  0
less_in_ga(x1, x2)  =  less_in_ga(x1)
less_out_ga(x1, x2)  =  less_out_ga
U7_ga(x1, x2, x3)  =  U7_ga(x3)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U4_AGA(x1, x2, x3, x4, x5, x6)  =  U4_AGA(x4, x6)

We have to consider all (P,R,Pi)-chains

(87) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(88) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DELETE_IN_AGA(tree(Y, Left, Right)) → U4_AGA(Right, less_in_ga(Y))
U4_AGA(Right, less_out_ga) → DELETE_IN_AGA(Right)

The TRS R consists of the following rules:

less_in_ga(0) → less_out_ga
less_in_ga(s(X)) → U7_ga(less_in_ga(X))
U7_ga(less_out_ga) → less_out_ga

The set Q consists of the following terms:

less_in_ga(x0)
U7_ga(x0)

We have to consider all (P,Q,R)-chains.

(89) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• U4_AGA(Right, less_out_ga) → DELETE_IN_AGA(Right)
  The graph contains the following edges 1 >= 1

• DELETE_IN_AGA(tree(Y, Left, Right)) → U4_AGA(Right, less_in_ga(Y))
  The graph contains the following edges 1 > 1

(90) TRUE